# christianp/PHD-Notes

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 @@ -714,9 +714,16 @@ \section{Computability of Bass-Serre groups \label{bass-serre}} sequence of the definition may not be immediately apparent.)} \end{proof} +\ajd{Before stating the next theorem the precise definition of \grz{n}-computable +homomorphism and \grz{n}-decidable subgroup, +should be given and discussed (see e.g. Cannonito\&Gatterdam '73).} \begin{theorem} \label{fgoagogcomp} -Suppose we have $G = \fgoagog$, where $\Gamma$ has $\nu$ vertices. Suppose all the $G_i$ are f.g. \grz{n}-computable groups for $n \geq 3$. Assume all the identified subgroups (\ajd{edge groups}) are \grz{n}-decidable, and all the isomorphisms $\phi_e$ are \grz{n}-computable. Then $G$ is \grz{n+1}-computable. +Suppose we have $G = \fgoagog$, where $\Gamma$ has $\nu$ vertices. Suppose +all the $G_i$ are f.g. \ajd{(finitely generated'': weed out abbreviations)} +\grz{n}-computable groups for $n \geq 3$. Assume all the identified subgroups +(\ajd{edge groups}) are \grz{n}-decidable, and all the isomorphisms $\phi_e$ +are \grz{n}-computable. Then $G$ is \grz{n+1}-computable. \end{theorem} \begin{proof}